Finite Group Databasebeta

← C67×#11756 #30691007

Subgroups — C67×#11756 #30691007

Order 17152.

444 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2599C211yesmaximal, characteristic
2C67×#1740C211yesmaximal, characteristic
3C67×#1464C211yesmaximal, characteristic
4C67×#1351C212yesmaximal
5C67×#1344C212yesmaximal
6C2×C4×Q8×C67C411yescharacteristic, complemented
7C2×#117×C67—21no—
8C2×#117×C67C2211yescharacteristic
9#432×C67—21no—
10#398×C67C411yescharacteristic, complemented
11#397×C67C2211yescharacteristic
12#395×C67C411yescharacteristic, complemented
13#393×C67C2211yescharacteristic
14#388×C67—21no—
15#387×C67—21no—
16#387×C67—21no—
17#385×C67—21no—
18#383×C67C411yescharacteristic, complemented
19#342×C67—21no—
20#342×C67C2212yes—
21#324×C67—21no—
22#323×C67C2212yes—
23C22×Q8×C67C2×C411yescharacteristic
24C23×C4×C67D411yescharacteristic
25C4×Q8×C67—41no—
26C4×Q8×C67—41no—
27#117×C67—24no—
28#117×C67—41no—
29#117×C67—41no—
30#117×C67—42no—
31C2×C4⋊C4×C67—21no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—41no—
35C2×C4⋊C4×C67C2×C411yescharacteristic
36C2×C4⋊C4×C67C2×C411yescharacteristic
37C2×C4⋊C4×C67D411yescharacteristic
38C2×C22⋊C4×C67—22no—
39C2×C22⋊C4×C67—21no—
40C2×C22⋊C4×C67—21no—
41C2×C22⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67C2311yescharacteristic
43C2×C42×C67—21no—
44C2×C42×C67—41no—
45C2×C42×C67C2×C411yescharacteristic
46C2×C42×C67D411yescharacteristic
47C4⋊C8×C67—41no—
48C22⋊C8×C67—22no—
49#97×C67—41no—
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67—21no—
56#95×C67C2×C411yescharacteristic
57#95×C67C2×C411yescharacteristic
58C24×C67C2×D411yescharacteristic
59C2×Q8×C67—42no—
60C2×Q8×C67—41no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—22no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—42no—
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22×C411yescharacteristic
76C22×C4×C67C2×D411yescharacteristic
77C2×C8×C67—42no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—42no—
82C4⋊C4×C67—44no—
83C4⋊C4×C67—41no—
84C4⋊C4×C67—41no—
85C4⋊C4×C67—42no—
86C22⋊C4×C67—24no—
87C22⋊C4×C67—41no—
88C22⋊C4×C67—41no—
89C22⋊C4×C67—42no—
90C22⋊C4×C67—41no—
91C22⋊C4×C67—42no—
92C22⋊C4×C67—41no—
93C42×C67—21no—
94C42×C67—21no—
95C42×C67—21no—
96C42×C67—21no—
97C42×C67—42no—
98C42×C67—44no—
99C23×C67—21no—
100C23×C67—21no—
101C23×C67—41no—
102C23×C67—41no—
103C23×C67C2×C22⋊C411yescharacteristic
104C23×C67AGL(1,ℤ/8)11yescharacteristic
105C23×C67#13711yescharacteristic
106Q8×C67—42no—
107Q8×C67—42no—
108C2×C4×C67—22no—
109C2×C4×C67—22no—
110C2×C4×C67—22no—
111C2×C4×C67—22no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—22no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—44no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—82no—
138C2×C4×C67—82no—
139C2×C4×C67#9911yescharacteristic
140C2×C4×C67#9911yescharacteristic
141C2×C4×C67#10412yes—
142C8×C67—82no—
143C22×C67—21no—
144C22×C67—21no—
145C22×C67—21no—
146C22×C67—21no—
147C22×C67—41no—
148C22×C67—41no—
149C22×C67—41no—
150C22×C67—41no—
151C22×C67—81no—
152C22×C67C2×#9911yescharacteristic
153C22×C67#41711yescharacteristic
154C22×C67C2×#10411yescharacteristic
155C4×C67—21no—
156C4×C67—21no—
157C4×C67—22no—
158C4×C67—41no—
159C4×C67—42no—
160C4×C67—41no—
161C4×C67—41no—
162C4×C67—41no—
163C4×C67—81no—
164C4×C67—81no—
165C4×C67—84no—
166C4×C67—81no—
167#11756C6711yesmaximal, characteristic, direct product factor, Sylow
168C2×C67—21no—
169C2×C67—21no—
170C2×C67—41no—
171C2×C67—41no—
172C2×C67#150411yescharacteristic
173C2×C67#212011yescharacteristic
174C2×C67#212911yescharacteristic
175#2599C2×C6711yescharacteristic
176#1740C2×C6711yescharacteristic
177#1464C2×C6711yescharacteristic
178#1351C2×C6712yes—
179#1344C2×C6712yes—
180C67#1175611yescharacteristic, direct product factor, Sylow
181C2×C4×Q8C4×C6711yescharacteristic, complemented
182C2×#117—21no—
183C2×#117C22×C6711yescharacteristic
184#432—21no—
185#398C4×C6711yescharacteristic, complemented
186#397C22×C6711yescharacteristic
187#395C4×C6711yescharacteristic, complemented
188#393C22×C6711yescharacteristic
189#388—21no—
190#387—21no—
191#387—21no—
192#385—21no—
193#383C4×C6711yescharacteristic, complemented
194#342—21no—
195#342C22×C6712yes—
196#324—21no—
197#323C22×C6712yes—
198C22×Q8C2×C4×C6711yescharacteristic
199C23×C4D4×C6711yescharacteristic
200C4×Q8—41no—
201C4×Q8—41no—
202#117—24no—
203#117—41no—
204#117—41no—
205#117—42no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C2×C4×C6711yescharacteristic
211C2×C4⋊C4C2×C4×C6711yescharacteristic
212C2×C4⋊C4D4×C6711yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4D4×C6711yescharacteristic
217C2×C22⋊C4C23×C6711yescharacteristic
218C2×C42—21no—
219C2×C42—41no—
220C2×C42C2×C4×C6711yescharacteristic
221C2×C42D4×C6711yescharacteristic
222C4⋊C8—41no—
223C22⋊C8—22no—
224#97—41no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95C2×C4×C6711yescharacteristic
232#95C2×C4×C6711yescharacteristic
233C24C2×D4×C6711yescharacteristic
234C2×Q8—42no—
235C2×Q8—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4C22⋊C4×C6711yescharacteristic
247C22×C4C22⋊C4×C6711yescharacteristic
248C22×C4C22⋊C4×C6711yescharacteristic
249C22×C4C22⋊C4×C6711yescharacteristic
250C22×C4C22×C4×C6711yescharacteristic
251C22×C4C2×D4×C6711yescharacteristic
252C2×C8—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C22⋊C4—24no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—42no—
273C42—44no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊C4×C6711yescharacteristic
279C23AGL(1,ℤ/8)×C6711yescharacteristic
280C23#137×C6711yescharacteristic
281Q8—42no—
282Q8—42no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4#99×C6711yescharacteristic
315C2×C4#99×C6711yescharacteristic
316C2×C4#104×C6712yes—
317C8—82no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×#99×C6711yescharacteristic
328C22#417×C6711yescharacteristic
329C22C2×#104×C6711yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—42no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—84no—
341C4—81no—
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2C67×#150411yescharacteristic
347C2C67×#212011yescharacteristic
348C2C67×#212911yescharacteristic
subgroupG/H
center, socleC22×C67C2×#99
commutatorC22×C4C22×C4×C67
Frattini, cosocleC2×C22⋊C4C23×C67
prime radicalC24×C67C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C22×C4×C67 ◁ C2×#117×C67 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. C2×C4×Q8×C67C4semidirect
10. #398×C67C4semidirect
12. #395×C67C4semidirect
18. #383×C67C4semidirect
167. #11756C67direct (N × H)
180. C67#11756direct (N × H)
181. C2×C4×Q8C4×C67semidirect
185. #398C4×C67semidirect
187. #395C4×C67semidirect
193. #383C4×C67semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C67×#25992
C67×#17402
C67×#14642
C67×#13512
C67×#13442
#1175667